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Anthropic reasoning is a critical tool to understand probabilities, especially in a large universe or multiverse. According to anthropic reasoning, we should consider ourselves typical among members of a reference class that must include…
We consider the problem of estimating the density $\Pi$ of a determinantal process $N$ from the observation of $n$ independent copies of it. We use an aggregation procedure based on robust testing to build our estimator. We establish…
Introductory statistical inference texts and courses treat the point estimation, hypothesis testing, and interval estimation problems separately, with primary emphasis on large-sample approximations. Here I present an alternative approach…
In this note, we revisit a classical problem related to the density of nonlinear statistics. We obtain a new representation of densities and, for the first time, a necessary and sufficient condition for the existence of densities is…
An alternative computational approach to the Collatz (3n+1) conjecture is presented that may be theoretically capable of confirming the conjecture.
We consider the near-critical Erd\H{o}s-R\'{e}nyi random graph $G(n,p)$ and provide a new probabilistic proof of the fact that, when $p$ is of the form $p=p(n)=1/n+\lambda/n^{4/3}$ and $A$ is large,…
The problem of guessing a random string is revisited. A close relation between guessing and compression is first established. Then it is shown that if the sequence of distributions of the information spectrum satisfies the large deviation…
In this pedagogical text aimed at those wanting to start thinking about or brush up on probabilistic inference, I review the rules by which probability distribution functions can (and cannot) be combined. I connect these rules to the…
We introduce $p$-derivations and give a few basic ways in which they act like derivatives by numbers.
We improve some results on the size of the greatest prime factor of integers of the form ab+1, where a and b belong to finite sets of integers with rather large density.
We present an improved incremental selection algorithm of the selection algorithm presented in [1] and prove all the selected conjectures.
We prove a multivariate version of Bernstein's inequality about the probability that degenerate $U$-statistics take a value larger than some number $u$. This is an improvement of former estimates for the same problem which yields an…
We extend the study by Ornstein and Weiss on the asymptotic behavior of the normalized version of recurrence times and establish the large deviation property for a certain class of mixing processes. Further, an estimator for entropy based…
Comment on ``Lancaster Probabilities and Gibbs Sampling'' [arXiv:0808.3852]
Recent decades have seen an interest in prediction problems for which Bayesian methodology has been used ubiquitously. Sampling from or approximating the posterior predictive distribution in a Bayesian model allows one to make inferential…
Consider a set of order statistics that arise from sorting samples from two different populations, each with their own, possibly different distribution function. The probability that these order statistics fall in disjoint, ordered…
We consider partially observed multiscale diffusion models that are specified up to an unknown vector parameter. We establish for a very general class of test functions that the filter of the original model converges to a filter of reduced…
A note on "Bayesian nonparametric estimators derived from conditional Gibbs structures" by Antonio Lijoi, Igor Pr\"{u}nster, Stephen G. Walker [arXiv:0808.2863].
When a real-valued function of one variable is approximated by its $n^{th}$ degree Taylor polynomial, the remainder is estimated using the Alexiewicz and Lebesgue $p$-norms in cases where $f^{(n)}$ or $f^{(n+1)}$ are Henstock--Kurzweil…
We respond to comments on our paper, titled "Instrumental variable estimation of the causal hazard ratio."